Episodes in the History of Modern Algebra (1800-1950)

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Jeremy J. Gray, Karen Hunger Parshall
American Mathematical Soc., Aug 31, 2011 - Mathematics - 336 pages
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Algebra, as a subdiscipline of mathematics, arguably has a history going back some 4000 years to ancient Mesopotamia. The history, however, of what is recognized today as high school algebra is much shorter, extending back to the sixteenth century, while the history of what practicing mathematicians call "modern algebra" is even shorter still. The present volume provides a glimpse into the complicated and often convoluted history of this latter conception of algebra by juxtaposing twelve episodes in the evolution of modern algebra from the early nineteenth-century work of Charles Babbage on functional equations to Alexandre Grothendieck's mid-twentieth-century metaphor of a ``rising sea'' in his categorical approach to algebraic geometry. In addition to considering the technical development of various aspects of algebraic thought, the historians of modern algebra whose work is united in this volume explore such themes as the changing aims and organization of the subject as well as the often complex lines of mathematical communication within and across national boundaries. Among the specific algebraic ideas considered are the concept of divisibility and the introduction of non-commutative algebras into the study of number theory and the emergence of algebraic geometry in the twentieth century. The resulting volume is essential reading for anyone interested in the history of modern mathematics in general and modern algebra in particular. It will be of particular interest to mathematicians and historians of mathematics.
 

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Contents

Acknowledgments
1
Functional Equations Language
13
Babbage and a Language for the Solution of Functional Equations
19
Babbage and FirstOrder Functional Equations in One Variable
26
De Gérandos Theory of Signs and Its Mathematical Implications
33
Destutt de Tracys Views on Artificial Languages
39
Duncan F Gregory the Calculus
49
14
71
49
89
Kroneckers Fundamental Theorem of General Arithmetic
107
On The Beginnings of
153
Leonard
179
Emmy Noethers 1932 ICM Lecture on Noncommutative Methods
199
Changing
221
A Historical Sketch of B L van der Waerdens Work
245
On the Arithmetization of Algebraic Geometry
285

Divisibility Theories in the Early History of Commutative Algebra
73
17
77
33
83
Grothendieck on Simplicity and Generality
301
Index
327
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